Mohr Circle Interactive Calculator

The Mohr circle is a graphical representation of stress transformation at a point, converting complex three-dimensional stress states into an intuitive two-dimensional visualization. Used extensively in structural analysis, geotechnical engineering, and mechanical design, this method allows engineers to determine principal stresses, maximum shear stress, and stress states at any orientation without repeated trigonometric calculations.

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Stress Element Diagram

Interactive Mohr Circle Calculator

Governing Equations

Theory & Practical Applications

Fundamentals of Stress Transformation

Mohr's circle, developed by Christian Otto Mohr in 1882, provides a graphical method for analyzing plane stress at a point. The fundamental principle rests on the invariance of certain stress combinations under coordinate rotation. While the individual stress components σ_x, σ_y, and τ_xy change with coordinate system orientation, the sum σ_x + σ_y remains constant—this is the first stress invariant. The circle's center represents this average normal stress, while its radius quantifies the stress deviation available for transformation.

The power of Mohr's circle lies in converting a trigonometric problem into a geometric one. Every point on the circle represents a valid stress state at some orientation angle θ. The critical insight that eludes many practitioners: on the physical stress element, angles rotate by θ, but on Mohr's circle, the corresponding point rotates by 2θ. This doubling occurs because the stress transformation equations contain terms like cos(2θ) and sin(2θ), directly linking physical rotation to graphical representation.

Construction Methodology and Sign Conventions

Proper construction begins with establishing sign conventions. The most widely adopted convention plots normal stresses along the horizontal axis (tensile positive to the right, compressive negative to the left) and shear stresses along the vertical axis. For shear stress, the convention that produces correct principal angle predictions assigns positive values to shear stresses that produce clockwise moments about the stress element's center, plotting them downward. This counterintuitive downward-positive convention for the vertical axis ensures that counterclockwise physical rotations correspond to counterclockwise movements on the circle.

To construct the circle: plot point A at coordinates (σ_x, τ_xy) and point B at (σ_y, -τ_xy). These two points represent stresses on perpendicular faces of the element. The line connecting A and B passes through the circle's center and forms a diameter. The center location C is at ((σ_x + σ_y)/2, 0), and the radius equals the distance from C to either A or B. Principal stresses occur where the circle intersects the horizontal axis—maximum at the rightmost intersection (σ₁) and minimum at the leftmost (σ₂). Maximum shear stress equals the circle radius, occurring at the top and bottom extremes where the element has no normal stress differential.

Critical Engineering Limitations

Mohr's circle applies strictly to plane stress conditions where one principal stress is zero (or negligibly small). This assumption holds for thin-walled structures, surface conditions, and situations where the out-of-plane dimension is small relative to in-plane dimensions. For fully three-dimensional stress states, three separate Mohr circles are required—one for each pair of principal stresses. The largest circle (connecting σ₁ and σ₃) governs maximum shear stress, which can be significantly larger than the maximum in-plane shear stress predicted by a single circle analysis. This distinction matters critically in thick-walled pressure vessels and deep geological structures where σ_z cannot be neglected.

Material failure theories interact differently with Mohr's circle predictions. Brittle materials following maximum normal stress theory fail when σ₁ exceeds ultimate tensile strength, directly readable from the circle. Ductile materials following von Mises or Tresca criteria require additional calculations—the circle provides input values but doesn't directly predict yielding. The Tresca criterion states yielding occurs when maximum shear stress reaches σ_y/2, where σ_y is yield strength. This converts circle radius into a failure metric: R_max = σ_y/2. The von Mises criterion, while more accurate, requires computing (σ₁² + σ₂² - σ₁σ₂)^1/2 and comparing to yield strength, using principal stresses extracted from the circle.

Industrial Applications Across Disciplines

In geotechnical engineering, Mohr's circle combines with the Mohr-Coulomb failure criterion to analyze soil stability. Cohesive soils exhibit a failure envelope defined by τ_f = c + σ tan(φ), where c is cohesion and φ is friction angle. The Mohr circle for in-situ stress must not touch this envelope or failure occurs. For a vertical excavation in clay with σ_v = 150 kPa vertical stress, σ_h = 75 kPa horizontal stress (K₀ = 0.5), and τ = 0 initially, the principal stresses are already aligned with coordinate axes. Excavation releases horizontal stress on one face, creating σ_x = 0 and σ_y = 150 kPa with τ_xy = 0. The new circle has center at 75 kPa and radius 75 kPa. If soil cohesion c = 40 kPa and φ = 20°, the failure envelope intersects at τ = 40 + σ·tan(20°) = 40 + 0.364σ. The circle will touch this envelope when radius equals the perpendicular distance from center to envelope, predicting instability.

Pressure vessel analysis requires Mohr's circle for combined loading. Consider a thin-walled cylindrical tank with 1.2 m diameter, 8 mm wall thickness, under 2.5 MPa internal pressure. Hoop stress σ_θ = pr/t = 2.5 × 0.6 / 0.008 = 187.5 MPa. Longitudinal stress σ_L = pr/(2t) = 93.75 MPa. For an element on the cylinder surface with no external shear, the stress state is σ_x = 187.5 MPa (hoop), σ_y = 93.75 MPa (longitudinal), τ_xy = 0. Mohr's circle center: (187.5 + 93.75)/2 = 140.625 MPa. Radius: (187.5 - 93.75)/2 = 46.875 MPa. Principal stresses: σ₁ = 187.5 MPa, σ₂ = 93.75 MPa, σ₃ = 0 (radial, negligible for thin wall). Maximum shear stress τ_max = (σ₁ - σ₃)/2 = 93.75 MPa, occurring on a 45° plane through the wall thickness. The in-plane maximum shear stress of 46.875 MPa significantly underestimates the true maximum.

Worked Example: Shaft Under Combined Torsion and Bending

A solid circular shaft with 50 mm diameter experiences a bending moment M = 1.8 kN·m and torque T = 1.2 kN·m. At a critical surface point on the tension side, determine principal stresses, maximum shear stress, and their orientations.

Step 1: Calculate stress components
Section modulus: S = πd³/32 = π(0.05)³/32 = 1.534 × 10⁻⁵ m³
Polar moment: J = πd⁴/32 = π(0.05)⁴/32 = 6.136 × 10⁻⁷ m⁴
Bending stress: σ_x = M/S = 1800 / (1.534 × 10⁻⁵) = 117.35 MPa (tension)
Torsional shear: τ_xy = Tr/J = 1200 × 0.025 / (6.136 × 10⁻⁷) = 48.90 MPa
Normal stress perpendicular to bending: σ_y = 0 MPa

Step 2: Calculate Mohr circle parameters
Center: σ_avg = (117.35 + 0)/2 = 58.675 MPa
Radius: R = √[((117.35 - 0)/2)² + 48.90²] = √[58.675² + 48.90²] = √(3442.76 + 2391.21) = √5833.97 = 76.38 MPa

Step 3: Determine principal stresses
σ₁ = σ_avg + R = 58.675 + 76.38 = 135.06 MPa
σ₂ = σ_avg - R = 58.675 - 76.38 = -17.70 MPa (compression)
Principal angle: tan(2θ_p) = 2τ_xy/(σ_x - σ_y) = 2(48.90)/(117.35 - 0) = 0.8333
2θ_p = arctan(0.8333) = 39.81°
θ_p = 19.91° counterclockwise from longitudinal axis

Step 4: Maximum shear stress
τ_max = R = 76.38 MPa
Occurs at θ_s = θ_p + 45° = 19.91° + 45° = 64.91° from longitudinal axis
Normal stress at max shear plane: σ_n = σ_avg = 58.675 MPa

Engineering significance: The maximum principal stress (135.06 MPa) exceeds the bending stress alone (117.35 MPa) by 15%, demonstrating how combined loading amplifies peak stresses. For design using the Tresca criterion with a yield strength of 250 MPa, the safety factor is SF = σ_y/(2τ_max) = 250/(2 × 76.38) = 1.64. This relatively low factor indicates the shaft is operating near capacity. Additionally, the principal stress plane at 19.91° doesn't align with the longitudinal axis, meaning failure initiation would occur on an inclined plane—critical for predicting crack propagation paths.

Advanced Considerations in Pole Method

The pole method provides an elegant graphical technique for finding stress on any plane without calculations. Once the Mohr circle is constructed, locate the "pole point" P by drawing a line from point A (σ_x, τ_xy) parallel to the physical x-face. Where this line intersects the circle defines the pole. Subsequently, any line drawn from P through the circle automatically represents stress on a plane parallel to that line in physical space. This method proves invaluable when visualizing multiple plane orientations simultaneously or when working graphically without computational aids.

The pole method reveals a subtle but powerful concept: the pole represents the unique point from which all radius lines correspond directly to physical plane orientations without angle doubling. While standard Mohr circle interpretation requires converting 2θ on the circle to θ in physical space, lines through the pole maintain direct angular correspondence. This distinction becomes crucial in complex analysis involving multiple stress states or iterative design adjustments.

Additional Engineering Insights

For quality assurance, verify that the sum σ_x + σ_y equals σ₁ + σ₂—this stress invariant must hold regardless of calculation errors. Also verify that points 180° apart on the circle (representing perpendicular planes in physical space) have shear stresses equal in magnitude but opposite in sign, confirming equilibrium. When τ_xy = 0, the given coordinate system already aligns with principal directions, producing a degenerate case where circle construction is unnecessary but instructive for verification. Explore the interactive calculator at FIRGELLI's engineering calculator hub for additional stress analysis tools including von Mises stress, Tresca criterion, and combined loading scenarios.

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